| L(s) = 1 | − 5-s + 7-s − 3·11-s − 13-s + 3·17-s + 2·19-s + 6·23-s + 25-s + 9·29-s + 8·31-s − 35-s − 10·37-s + 2·43-s + 3·47-s + 49-s + 3·55-s − 12·59-s + 8·61-s + 65-s + 8·67-s + 14·73-s − 3·77-s + 5·79-s + 12·83-s − 3·85-s − 12·89-s − 91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s + 1.43·31-s − 0.169·35-s − 1.64·37-s + 0.304·43-s + 0.437·47-s + 1/7·49-s + 0.404·55-s − 1.56·59-s + 1.02·61-s + 0.124·65-s + 0.977·67-s + 1.63·73-s − 0.341·77-s + 0.562·79-s + 1.31·83-s − 0.325·85-s − 1.27·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.539540605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539540605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.798799637566292881800742343892, −8.716629455833952794282379017179, −8.064936832254552873896534622622, −7.34350328382360783706346369628, −6.46860294873273970390217696172, −5.22587228095951654792844133362, −4.76452746231177699719724258883, −3.44205825088523197919741451661, −2.55483991183069062380192314926, −0.943810287462265257086373836609,
0.943810287462265257086373836609, 2.55483991183069062380192314926, 3.44205825088523197919741451661, 4.76452746231177699719724258883, 5.22587228095951654792844133362, 6.46860294873273970390217696172, 7.34350328382360783706346369628, 8.064936832254552873896534622622, 8.716629455833952794282379017179, 9.798799637566292881800742343892
